POVM

In functional analysis and quantum measurement theory, a positive-operator valued measure (POVM) is a measure whose values are non-negative self-adjoint operators on a Hilbert space, and whose integral is the identity operator. It is the most general formulation of measurements in quantum physics. Since projective measurements on a large system—i.e., measurements that are performed mathematically by a projection-valued measure (PVM)—will act on a subsystem in ways that cannot be described by a PVM on the subsystem alone, the POVM formalism becomes necessary. POVMs are used in the field of quantum information. In functional analysis and quantum measurement theory, a positive-operator valued measure (POVM) is a measure whose values are non-negative self-adjoint operators on a Hilbert space, and whose integral is the identity operator. It is the most general formulation of measurements in quantum physics. Since projective measurements on a large system—i.e., measurements that are performed mathematically by a projection-valued measure (PVM)—will act on a subsystem in ways that cannot be described by a PVM on the subsystem alone, the POVM formalism becomes necessary. POVMs are used in the field of quantum information. In rough analogy, a POVM is to a PVM what a density matrix is to a pure state. Density matrices are needed to specify the state of a subsystem of a larger system, even when the latter is in a pure state (see purification of quantum state); analogously, POVMs on a physical system are used to describe the effect of a projective measurement performed on a larger system. Historically, the term probability-operator measure (POM) has been used as a synonym for POVM, although this usage is now rare. In quantum field theory projective measurements are ill defined and lead to many contradictions. In the simplest case, a POVM is a set of Hermitian positive semidefinite operators { F i } {displaystyle {F_{i}}} on a Hilbert space H {displaystyle {mathcal {H}}} that sum to the identity operator, This formula is a generalization of the decomposition of a (finite-dimensional) Hilbert space by a set of orthogonal projectors, { E i } {displaystyle {E_{i}}} , defined for an orthogonal basis { | ϕ i ⟩ } {displaystyle {left|phi _{i} ight angle }} by: An important difference is that the elements of a POVM are not necessarily orthogonal, with the consequence that the number of elements in the POVM, n, can be larger than the dimension, N, of the Hilbert space they act in.